Rayleigh distribution and its generalizations

Abstract

Department of Ele ctric al Enginee ring, University of C olor ado, Boulde r, C olo.(R eceived D eccmber 7, ] 963) Th e Rayleigh d istribu tion is the di stribution of thc sum of a large num be r of coplanar (or t im c) vectors \\'ith r andom. amplitudes alld Ull ifO rlllly distributed phases. As s uch, it is the lim iti ng case of di stributions a~soci at('d II ith 111 0re gl'IlPral v pctor surn s t hat arise i n practical problems. S uch cases arC' til(' folloll' ing: (a) 'I'll(' p hasC' distribu tions of the vec tor tC'rms are not uni fo rm , e.g., in the case of scatt l' ring fro m rough surface'S; (b) One or more vector tcrms predominaLP, their nwan sq uare va lue' not I>l'illg neglig;ble co m pared to th e m ean square valup of Uw SU Ill , c.g., in tJle ea~e of sigl als propagat ed in eil i (,'. ll1<'teor-scatter, and atmosplwrie noi se; (0) Th e numiJer of \'('ctO I' t('r n ' j~ small, ('.g., in rada r ret ur ns from several clo",(' targets; (d) '1'110 Jlumi)('r of vector le l'm o is ilsl'H randolll , e.g., in at mosp her ic t u rbuh'nce, m ekor-scatter and atm ~pl'(' .. ic noise'. The ('('suI ting distributio ns for t hese cases and their deviatio ns from the R ayleig h distr ibution will be co nsidered.

Observations of multi-microsecond VHF pulsetrains in energetic intracloud lightning discharges

“…Note the logarithmic ordinate scale. Also shown is a thin straight line indicating a Rayleigh distribution of the electric-field envelope, in which ERP/<ERP> would be expected to be an exponential (Beckman, 1964). The Rayleigh distribution would be expected if the signal were caused by random fading due to a summation of numerous uncorrelated inputs.…”

The Distribution of Returns

“…The Weibull is an extreme value distribution and under the Rayleigh parameterization implies that large values are probable. As the Rayleigh distribution, it is the limiting distribu-tion of the vector sum with "random amplitudes and uniformly distributed phases" [30]. Through the χ distribution it is the distribution of the square root of the sum of squares of independent draws from a normal distribution.…”

A mathematical analysis of cellular interference on the performance of S-band military radar systems

“…This assumption is justified since radar returns from a target are composed of numerous and diverse reflecting elements. Then, statistically, the sum of these large number of independent random vectors, each having a uniform phase and a Rayleigh amplitude distribution, is a Rayleigh vector [18]. However, the Rayleigh assumption is generally true for targets having small radar cross section (RCS) because targets with high RCS exhibit a heavier tail than the Rayleigh distribution [19].…”

“…Then, the bound on probability of detection P D can be evaluated from equation (18) by using the result of equation (19).…”